Answer:
x= -
2
/3 = −27 = 2
/3−3i =3 =1.5000−2.5981i
x= 2
/3+ −27
= 2
/3+3i/3 = 1.5000+2.5981i
x=−3
Step by Step Explanation:
5.6 Solving x2-3x+9 = 0 by the Quadratic Formula .
According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
x = ————————
2A
In our case, A = 1
B = -3
C = 9
Accordingly, B2 - 4AC =
9 - 36 =
-27
Applying the quadratic formula :
3 ± √ -27
x = —————
2
In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)
Both i and -i are the square roots of minus 1
Accordingly,√ -27 =
√ 27 • (-1) =
√ 27 • √ -1 =
± √ 27 • i
Can √ 27 be simplified ?
Yes! The prime factorization of 27 is
3•3•3
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 27 = √ 3•3•3 =
± 3 • √ 3
√ 3 , rounded to 4 decimal digits, is 1.7321
So now we are looking at:
x = ( 3 ± 3 • 1.732 i ) / 2
Two imaginary solutions :
x =(3+√-27)/2=(3+3i√ 3 )/2= 1.5000+2.5981i
or:
x =(3-√-27)/2=(3-3i√ 3 )/2= 1.5000-2.5981i
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